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Phase diagram of insulating crystal and quantum Hall states in ABC-stacked trilayer graphene

In the presence of a perpendicular magnetic field, ABC-stacked trilayer graphene's chiral band structure supports a 12-fold degenerate N=0 Landau level (LL). Along with the valley and spin degrees of freedom, the zeroth LL contains additional quantum numbers associated with the LL orbital index $% n=0,1,2$. Remote inter-layer hopping terms and external potential difference $Δ_{B}$ between the layers lead to LL splitting by introducing a gap $% Δ_{LL}$ between the degenerate zero-energy triplet LL orbitals. Assuming that the spin and valley degrees of freedom are frozen, we study the phase diagram of this system resulting from competition of the single particle LL splitting and Coulomb interactions within the Hartree-Fock approximation at integer filling factors. Above a critical value $Δ_{LL}^{c}$ of the external potential difference i,e, for $|Δ_{LL}| >Δ_{LL}^{c}$, the ground state is a uniform quantum Hall state where the electrons occupy the lowest unoccupied LL orbital index. For $|Δ_{LL}| <Δ_{LL}^{c}$ (which corresponds to large positive or negative values of $Δ_{B}$) the uniform QH state is unstable to the formation of a crystal state at integer filling factors. This phase transition should be characterized by a Hall plateau transition as a function of $Δ_{LL}$ at a fixed filling factor. We also study the properties of this crystal state and discuss its experimental detection.